More by Francesco Petitta

More by Sergio Segura de León

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Abstract

In this paper we show approximation procedures for studying singular elliptic
problems whose model is $$ \begin{cases} - \Delta u= b(u)|\nabla u|^2+f(x),
& \text{in } \; \Omega;\\ u = 0, & \text{on } \; \partial \Omega;
\end{cases} $$ where $b(u)$ is singular in the $u$-variable at $u=0$, and $f \in
L^m (\Omega)$, with $m>\frac N2$, is a function that does not have a constant
sign. We will give an overview of the landscape that occurs when different
problems (classified according to the sign of $b(s)$) are considered. So, in
each case and using different methods, we will obtain a priori estimates, prove
the convergence of the approximate solutions, and show some regularity
properties of the limit.